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Differentiable semigroups are Lie groups. (English) Zbl 0836.22005
Summary: We present here a modern, detailed proof to the following theorem which was introduced by G. Birkhoff [Trans. Am. Math. Soc. 43, 61-101 (1938; Zbl 0018.20502)] in 1938: If \(S\) is a local semigroup with neighborhood of 1 homeomorphic to a Banach space and with multiplication strongly differentiable at 1, then \(S\) is a local Lie group. – Although this theorem is more than 50 years old and remains the strongest result relating to Hilbert’s fifth problem in the infinite dimensional setting, it is frequently overlooked in favor of weaker results. Therefore, it is the goal of the authors here to clarify its importance and to demonstrate a proof which is more accessible to contemporary readers than the one offered by Birkhoff.
MSC:
22A20 Analysis on topological semigroups
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